STEP 1 - Tap on "Room" from the navigation menu at the bottom of your screen.
OPTIONAL: While in the Dorm Tower, tap on "My Room" on the ground floor

STEP 2 - Once you are on the "Room" page, tap on the furniture slot you wish to change.

STEP 3 - Tap on "Owned" tab.
        You will should see two tabs labeled "Shop" and "Owned"

STEP 4 - Tap on the furniture piece you wish to equip.
        All owned furniture pieces for that furniture slot will be displayed here

STEP 5 - Your room should update to the furniture that you chose!